Question

Solve system of equations using Cramer’s Rule.

$\left\{\begin{array}{l}5x-3y=-1\\ 2x-y=2\end{array}\right.$

Short answer

The solution for the system of linear equation is$(7,12)$

Step-by-step solution

Step 1. Given information

The system of linear equations is$\left\{\begin{array}{l}5x-3y=-1\\ 2x-y=2\end{array}\right.$

Step 2. Find the determinant D and evaluate it   

We will take the coefficients of the variables to form the determinant $D$

$D=\left|\begin{array}{cc}5& -3\\ 2& -1\end{array}\right|$

Evaluate it,

$$\begin{gathered} D=\left(5\right)(-1)-(-3)\left(2\right) \\ D=-5+6 \\ D=1 \end{gathered}$$

Step 3. Find the determinant Dx and evaluate it

We will use the constants in place of the $x$coefficients to find the determinant $D_x$

$D_x=\left|\begin{array}{cc}-1& -3\\ 2& -1\end{array}\right|$

Evaluate it

$$\begin{gathered} D_x=(-1)(-1)-(-3)\left(2\right) \\ {D}_x=1+6 \\ {D}_x=7 \end{gathered}$$

Step 4. Find the determinant Dy and evaluate it

We will use the constants in place of the $y$coefficients to find the determinant $D_y$

$D_y=\left|\begin{array}{cc}5& -1\\ 2& 2\end{array}\right|$

Evaluate it

$$\begin{gathered} D_y=\left(5\right)\left(2\right)-(-1)\left(2\right) \\ {D}_y=10+2 \\ {D}_y=12 \end{gathered}$$

Step 5. Find x and y

To find $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{7}{1} \\ x=7 \end{gathered}$$

To find $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{12}{1} \\ y=12 \end{gathered}$$

The solution for the system of linear equation is$(7,12)$

Step 6. Check by substituting values

Substitute $x=7.y=12$in both the equations,

$$\begin{gathered} 5x-3y=-1 \\ 5\left(7\right)-3\left(12\right)=-1 \\ 35-36=-1 \\ -1=-1 \end{gathered}$$

This is true.

$$\begin{gathered} 2x-y=2 \\ 2\left(7\right)-12=2 \\ 14-12=2 \\ 2=2 \end{gathered}$$

This is also true.

Therefore, the solution for the system of linear equation is $(7,12)$